It remains to show the fineness of the moduli space \(\widetilde{M}^{s}_{\mathcal {X}/S}(\mathcal {F}_{0},P,\delta )\) . By the argument in [[1]}, Lemmas 4.21, 4.22 and Remark 4.23] for each geometric point of \(S\) , one can show the universal family
\((\check{\mathcal {F}}_{\mathcal {X}\times _{S}\widetilde{\mathcal {R}}^{s}},\check{\varphi }_{\mathcal {X}\times _{S}\widetilde{\mathcal {R}}^{s}})\)  of \(\delta \) -stable pairs on \(\mathcal {X}/S\)  is \(\mathrm {PGL}(V)\) -linearized. It follows from [[1]}, Lemma 4.21] that the stabilizer group of a closed point of \(\widetilde{\mathcal {R}}^s\)  in \(\mathrm {PGL}(V)\)  is trivial due to [[1]}, Lemma 2.28] for every geometric point. By [[4]}, Theorem 4.2.12 and Corollary 4.2.13], the map \(\widetilde{\mathcal {R}}^s\rightarrow \widetilde{M}^{s}_{\mathcal {X}/S}(\mathcal {F}_{0},P,\delta )\)  is a principal \(\mathrm {PGL}(V)\) -bundle (or a \(\mathrm {PGL}(V)\) -torsor). Then \((\check{\mathcal {F}}_{\mathcal {X}\times _{S}\widetilde{\mathcal {R}}^{s}},\check{\varphi }_{\mathcal {X}\times _{S}\widetilde{\mathcal {R}}^{s}})\)  can be descended to be a universal family parameterized by \(\widetilde{M}^{s}_{\mathcal {X}/S}(\mathcal {F}_{0},P,\delta )\) , which completes the proof.
